How to calculate the resistance of a series circuit

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A series circuit is an electrical circuit where components are connected in a single path, one after another. In this type of circuit, there is only one path for current to flow through all components sequentially. Let's build a basic series circuit with a battery and three resistors. Notice how the current flows through each component in order, following the single available path. This is the fundamental characteristic of series circuits - if any component fails, the entire circuit stops working because there's no alternative path for current flow. The fundamental principle of series circuits is that current is the same at all points throughout the circuit. This happens because there is only one path for current to flow, and current cannot accumulate anywhere in the circuit. What goes in must come out. Let's demonstrate this with current meters placed at different positions in our series circuit. Notice that all three ammeters show identical readings of 2 amperes. The flowing particles represent current moving through the circuit, maintaining the same rate at every point. Mathematically, we express this as I1 equals I2 equals I3 equals I total. The voltage division principle states that voltage drops across individual resistors add up to the total voltage supplied by the battery. This is based on conservation of energy and Kirchhoff's voltage law. Each resistor consumes part of the total voltage as current flows through it. Let's demonstrate this with voltmeters connected across each component. The battery supplies 12 volts total. Notice how the first resistor drops 3 volts, the second resistor drops 4 volts, and the third resistor drops 5 volts. When we add these individual voltage drops together: 3 plus 4 plus 5 equals 12 volts, which matches our battery voltage exactly. Now let's derive the formula for total resistance in a series circuit step by step. We start with Ohm's law, which states that voltage equals current times resistance for each resistor. For the total circuit, the total voltage equals current times total resistance. From our voltage division principle, we know that total voltage equals the sum of individual voltage drops. Substituting Ohm's law for each resistor, we get current times total resistance equals current times R1 plus current times R2 plus current times R3. We can factor out the common current I from the right side. Finally, dividing both sides by current I, we arrive at our fundamental formula: total resistance equals R1 plus R2 plus R3. This shows that resistances simply add together in series circuits. Let's work through a specific example to demonstrate how to calculate total resistance in a series circuit. We have three resistors with values of 10 ohms, 20 ohms, and 30 ohms connected in series with a 12-volt battery. Using our derived formula, the total resistance equals R1 plus R2 plus R3. Substituting our values: total resistance equals 10 plus 20 plus 30, which equals 60 ohms. Now we can calculate additional circuit parameters. The total current using Ohm's law is 12 volts divided by 60 ohms, which equals 0.2 amperes. We can also find individual voltage drops: V1 equals 0.2 times 10 equals 2 volts, V2 equals 0.2 times 20 equals 4 volts, and V3 equals 0.2 times 30 equals 6 volts. Notice that these voltage drops add up to 12 volts, confirming our calculations.